436 Suggestions

Appetizers and Lessons for Mathematics and Reason ( Français)
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Mathematics 436

The course consists of the following three topics with continued emphasis on problem solving solving techniques.

Algebra: functions (changes in parameters, properties of graphs); polynomials (5 operation, factoring, rational expressions), graphing (linear and quadratic functions, sum, difference and product); systems of linear and quadratic equations; analytic geometry of the straight line.
Geometry: Simple proofs involving similar and congruent two- and three-dimensional figures; properties of three dimensional figures; trigonometric ratios, law of sines and law of cosines.
Statistics: Sampling Methods: methods (measures?) of central tendency, position and dispersion; quintile, quartile and percentile rank; box and whisker plots.

The above comes from The Lester B. Pearson School Board, Curriculum Selection Booklet, 2001-2. The latter had no part in the composition of page contents.

I would suggest the government of Quebec withdraw its approval and encourage the translation of the following works

Louise Lafortune et al, Mathematique 436, Collection Mathophilie, Tome 1 et 2, Guerin, Montreal Quebec, 514 842 3481 - Cost for schools: 34 CDN or less

for the use of English schools in Quebec. These two French language tomes offer clear, readable and logical development of mathematics 436. I recommend acquiring and reading these tomes to students in English school system, those who can read French.

I do not know if mathematics instructors in English schools can order copies of Mathematique 436, Collection Mathophilie, Tome 1 et 2 for use in course where some or all students have a good mastery of French. The latter might occur in schools where students are grouped or streamed according to language skills in English and French.

The identification of functions in Mathematique 436, Collection Mathophilie, Tome 1, with sets of ordered pairs, their graphs, is rather abrupt and without context - a problem in other well-written textbooks as well. The introduction of functions at this site offers a less abrupt route.

The approved pair of English language textbooks written by Guy Breton et al. for mathematics 436 is incoherent. For example, the word define appears in Book 1, while the discussion of what is a definition appears only in Book 2. Moreover many or most key words and concepts appear in bold-face type, but are not clearly defined. It appears that some concepts are out of sequence and others appears in name only. To see a clear and better model for the development of mathematical skills and concepts, one that a mathematician can appreciate in all or part, see Mathematique 436, Collection Mathophilie, Tome 1 et 2

Site lesson plans for Secondary I and Secondary II offer excellent preparation for senior high school mathematics in Quebec. Solving linear equations in one and two unknowns is a required part of the course. That part is recommended first (follow site lessons) in order to review and consolidate and force or encourage exact arithmetic skills with integers and rationals.

Students should aim to master numerical and algebraic methods for solving and understanding problems - watch for the forward and backward use of equations, and the literal or algebraic solution of equations. That forward and backward of equations, a skill named and emphasized here, is literally part of the old Quebec program for the development of algebraic thinking, a part that can be recognized in the intermediate objectives for secondary I, II and III courses. Items 1, 2 and 3 point to a more effective path for building skills and confidence in algebra, and thus providing a firmer foundation for the rest of secondary school mathematics.

The coverage of analytic geometry of straight lines, the coverage of exponents, the coverage of quadratics, the coverage of polynomials and the coverage of functions are essentially independent. So students and teachers may go over these items in any order they wish. The fourth topic on Functions could come after easier material on straight lines and polynomials, and before harder material on quadratics and exponents.

The site treatment of exponents is clear, precise and complete because it is based on properties of the natural logarithm and exponential functions (Secondary V level material in Quebec). Teachers will have to decide which is better. The treatment here or the treatment in the approved treatment. Site treatment is essentially full and correct, mathematically speaking, and if done now, will not have to be refined or redone later.

The site treatment of quadratics includes a step by step algebraic derivation of the quadratic formula. The derivation is preceded by numerical and algebraic examples that provide a model and a context for the algebraic steps. Mastery of the derivation is a good sign of competence in the algebraic shorthand way of writing and reasoning. Students should be encouraged to aim for and to struggle toward that mastery during the course or after. The intermediate objectives of the course require all the steps leading to the derivation, but not the derivation itself.

The site treatment of functions in site section on analytic geometry goes beyond secondary IV mathematics 436 including some secondary V material on inverse functions. But mathematics 436 students and their student should be interested in the context and motivation for the set-based view of function and relations in the site treatment of functions. We are trying remedy a situation where sets, functions and relations appear in secondary IV without explanation of the formal set-based codification.

The course development of trigonometry should begin in my opinion with the leanest account of similarity of triangles, so that students are not distracted or side-tracked by a long discourse on similarity.

If I could redesign the Mathematics 436 trig portion, I would start with the unit circle definition of trig functions with angles measured in degrees, and then show students how trig functions for acute angles can be computed from the ratio of sides of right triangles. That alternative start would provide a more logical base and a more efficient development for the appearance in this course of sines and cosines of obtuse angles. The alternative start described here actually occurs in the secondary V mathematics 536 but its appearance after the right-triangle introduction of trig functions for acute requires students to shift to the alternative route. Perhaps mathematics 436 should be taught the trig ratio approach along side a brief statement that an alternative way will be introduced in mathematics 536. (The alternative route is presently online in the analytic geometry site section).

Similarity itself is useful in making maps and plans in 2 and 3 dimensions. The interconnected proportionality relations which exist between real and image or model lengths, surfaces and volumes need to be treated separately and in context. More similarity appears in the cutting of lines by parallel lines to produce proportional line segments. That merits attention and proof, but not at length before the introduction of right triangle trig. Trig will appear in future studies while extensive discussion of similarity will almost surely disappear.

The site treatment of logic in mathematics begins with an introductory lesson comparing the different styles of thought and verification that appear. Implication and pattern based reason provides another means to develop and verify arithmetic, geometric and algebraic statements in mathematics alone or in its appearances outside of mathematics. The site treatment of proofs in geometry begins with the postulates (assumptions) of Euclidean Geometry that appear to be proven (derived from properties of rotations, translations and reflections) in the Quebec government approved textbooks, textbooks which do not clearly state their assumptions. While the site treatment of Euclidean geometry is self-contained and sufficient for most of the proofs seen in final examinations, in order to fulfill obligations of mathematics 436 instructor, I am hope to add a lesson or two in early 2007 to clarify matters.

The discussion of transformation geometry (dilatations, translations, rotations and reflections) begins in secondary II an continues through secondary III and now secondary IV. I suspect it disappears in secondary V. While this chain of reasons can lead to properties of transformations and hence an alternate base for proofs in geometry, see forthcoming site lessons, overall this transformation subprogram has no positive influence on the future studies of Quebec students. No Knowledge of it will be demanded in college level instruction.

Any one knowledgeable with spaghetti-type programming, knows that subprograms with no useful output can be eliminated from the program. Including transformation geometry in Quebec high schools while most students have difficulty with fractions and algebra distract studies and instruction from key and missing material. Talking about composite transformation in the plane or space months or years before students have met functions and function composition points to a lack of synchronization between algebra and geometry in the official high school program.

The English language textbook coverage of statistics includes unclear and unusual language. The Intermediate Objectives and French language textbooks such as

Louise Lafortune et al, Mathematique 436, Collection Mathophilie, Tome 1 et 2, Guerin, Montreal Quebec, 514 842 3481 - Cost for schools: 34 CDN or less

offer remedies for the lack of clarity. Final examinations include some strange backward use of percentile, quartile and quintile information to locate data values.

More Advice and Directions:

Many parts of the site areas

will help students prepare for and take mathematics 416, 426 and 436. Student in with a weak fraction sense and skills are inviting failure or substandard performance. Curious students may see a context for slopes alone or with polynomials in the first and second Calculus previews at this site.

The common introduct

Necessary Reminder

Students in the first year of high school may come with a weak to non-existence command of the times table (addition table too) and with a weak to non-existence fraction sense and abilities. The most important service of first year mathematics in high school is to consolidate fraction sense and skills. That is a prerequisite to algebra, geometry, trig and calculus. High school mathematics literally becomes a waste of time if fraction sense and skills are not consolidated and maintained in all years.